1. Columbus State Community College
Chapter 1 Section 8
Exponents and Order of Operations

2. Exponents and Order of Operations
Use exponents to write repeated factors.
Simplify expressions containing exponents.
Use the order of operations.
Simplify expressions with fraction bars.

3. Exponents
An exponent is a quick way to write repeated multiplication.
For example,
Exponent
3 • 3 • 3 • can be written
Base
This is called exponential notation or exponential form.
To simplify 34, actually do the multiplication.
34 = 3 • 3 • 3 • 3 = 81

4. Reading Common Exponents
Here are some examples of how to read common exponents.
is read “3 to the first power.”
is read “3 to the second power” or, more commonly, “3 squared.”
is read “3 to the third power” or, more commonly, “3 cubed.”
is read “3 to the fourth power.”
is read “3 to the fifth power.”

5. Using Exponents
EXAMPLE Using Exponents
Rewrite each multiplication using exponents. Also indicate how to read the exponential form.
(a)
10 • 10 • 10 • 10 • 10 • 10
can be written as 106, which is read “10 to the sixth power.”
(b)
( 7 )( 7 )
can be written as 72, which is read “7 squared” or “7 to the second power.”
(c) 2
can be written as 21, which is read “2 to the first power.”

6. Using Exponents with Negative Numbers
EXAMPLE Using Exponents with Negative Numbers
Simplify.
(a)
( –2 )3
= ( –2 ) ( –2 ) ( –2 )
4 ( –2 ) –8
(b)
( –2 )4
= ( –2 ) ( –2 ) ( –2 ) ( –2 )
– ( –2 ) From 2(a), ( –2 )3 = –8. 16

7. Using Exponents with Negative Numbers
EXAMPLE Using Exponents with Negative Numbers
Simplify.
( –5 ) ( –2 )3
(c)
( –5 )2 ( –2 )3
= ( –5 ) ( –5 ) ( –2 ) ( –2 ) ( –2 )
( 25 ) ( –8 )
–200

8. Calculator Tip – TI-30X IIS
Calculator Tip On your TI-30X IIS calculator, use the exponent key to enter exponents. To enter 75, press the following keys. ^ 7 ^ 5 =
16807 ^

9. yx Calculator Tip – TI-30Xa
Calculator Tip On your TI-30Xa calculator, use the exponent key to enter exponents. To enter 75, press the following keys. yx 7 yx 5 =
16807 yx

10. Working from Left to Right
EXAMPLE Working from Left to Right
Simplify.
(a)
–3 – –8 + –2
Do additions and subtractions from left to right. –2 3

11. Working from Left to Right
EXAMPLE Working from Left to Right
Simplify.
(b)
–20 ÷ 2 • 5
Do multiplications and divisions from left to right.
– • 5
–50

12. If we work from left to right
Mixing Operations
Compare the methods used to simplify the following example.
• 3
If we work from left to right
If we multiply first
• 3
• 3
• 3 36 16
Mathematicians have agreed to do things in a certain order. In this example, we multiply before we add.

13. Order of Operations
Order of Operations
Step 1 Work inside parentheses or other grouping symbols.
Step 2 Simplify expressions with exponents.
Step 3 Do the remaining multiplications and divisions as they occur from left to right.
Step 4 Do the remaining additions and subtractions as they occur from left to right.

14. CAUTION
CAUTION
To help in remembering the order of operations, you may have memorized the letters PEMDAS, or the phrase “Please Excuse My Dear Aunt Sally.”
Please Excuse My Dear Aunt Sally
1. Parentheses
2. Exponents
3. Multiply & Divide (from left to right)
4. Add & Subtract (from left to right)
Be careful! Do not automatically do all multiplication before division. Multiplication and division are done from left to right. Likewise, addition and subtraction are done from left to right.

15. Calculator Tip
Calculator Tip Enter the previous example in your calculator. 10 + 2 x 3 = 16
If you have a scientific calculator, it automatically uses the order of operations and multiplies first to get the correct answer of 16.

16. Using the Order of Operations with Whole Numbers
EXAMPLE Using the Order of Operations with Whole #’s
Simplify.
( 24 – 4 • 2 ) ÷ 8
Multiply inside parentheses first.
( 24 – ) ÷ 8
Subtract inside parentheses.
( ) ÷ 8
Multiply.
÷ 8
Divide.
Add. 9

17. Using the Order of Operations with Integers
EXAMPLE Using the Order of Operations with Integers
Simplify.
(a)
–40 ÷ ( 12 – 7 ) – 4
Subtract inside parentheses first.
–40 ÷ – 4
Divide.
– – 4
Subtract.
–12

18. Using the Order of Operations with Integers
EXAMPLE Using the Order of Operations with Integers
Simplify.
(b)
( 1 – –3 ) • ( 18 ÷ –9 )
Subtract inside parentheses first.
( ) • ( 18 ÷ –9 )
Divide inside parentheses.
( ) • ( – )
Multiply from left to right.
• ( – )
Multiply.
–40
Add.
–34

19. Using the Order of Operations with Exponents
EXAMPLE Using the Order of Operations with Exponents
Simplify.
(a)
( –5 )2 – ( –2 )3
Apply exponents.
– –8
Add. 33

20. Using the Order of Operations with Exponents
EXAMPLE Using the Order of Operations with Exponents
Simplify.
(b)
( –3 )3 – ( 8 – 6 )3 ( 3 )2
Subtract inside parentheses first.
( –3 )3 – ( )3 ( 3 )2
Apply exponents.
– – ( 9 )
Multiply.
– –
Add.
–99

21. Using the Order of Operations with Fraction Bars
EXAMPLE Using the Order of Operations with Fraction Bars
–4 – 2 ( 3 – –1 )2
3 • –4 ÷ 2 • –3
–36 18
Simplify. = = –2
First, do the work in the numerator
Next, do the work in the denominator
–4 – 2 ( 3 – –1 )2
3 • –4 ÷ 2 • –3
–4 – 2 ( )2
– ÷ 2 • –3
–4 – ( 16 )
– • –3
–4 – 18
–36

22. Exponents and Order of Operations
Chapter 1 Section 8 – End
Written by John T. Wallace